Showing papers on "Numerical analysis published in 1989"

On the limited memory BFGS method for large scale optimization

Abstract: We study the numerical performance of a limited memory quasi-Newton method for large scale optimization, which we call the L-BFGS method. We compare its performance with that of the method developed by Buckley and LeNir (1985), which combines cycles of BFGS steps and conjugate direction steps. Our numerical tests indicate that the L-BFGS method is faster than the method of Buckley and LeNir, and is better able to use additional storage to accelerate convergence. We show that the L-BFGS method can be greatly accelerated by means of a simple scaling. We then compare the L-BFGS method with the partitioned quasi-Newton method of Griewank and Toint (1982a). The results show that, for some problems, the partitioned quasi-Newton method is clearly superior to the L-BFGS method. However we find that for other problems the L-BFGS method is very competitive due to its low iteration cost. We also study the convergence properties of the L-BFGS method, and prove global convergence on uniformly convex problems.

Numerical methods and software

Abstract: 1. Introduction. 2. Computer Arithmetic and Computational Errors. 3. Linear systems of Equations. 4. Interpolation. 5. Numerical Quadrature. 6. Linear Least-Square Data Fitting. 7. Solution of Nonlinear Equations. 8. Ordinary Diernetial Equations. 9. Optimization and Nonlinear Least Squares. 10. Simulation and Random Numbers. 11. Trigonometirc Approximation and the Fast Fourier Transorm. Bibliography.

A class of high resolution explicit and implicit shock-capturing methods

Abstract: The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.

Convergence of Adomian's Method

Abstract: Adomian has developed a numerical technique using special kinds of polynomials for solving non‐linear functional equations. General conditions and a new formulation are proposed for proving the convergence of Adomian's method for the numerical resolution of non‐linear functional equations depending on one or several variables. The methods proposed are applicable to a very wide class of problems.

Interior path following primal-dual algorithms. Part I: Linear programming

Abstract: We describe a primal-dual interior point algorithm for linear programming problems which requires a total of $$O\left( {\sqrt n L} \right)$$ number of iterations, whereL is the input size. Each iteration updates a penalty parameter and finds the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem. The algorithm is based on the path following idea.

A posteriori error estimators for the Stokes equations

Abstract: We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.

Nonlinear Galerkin methods

Abstract: This article presents a new method of integrating evolution differential equations—the non-linear Galerkin method—that is well adapted to the long-term integration of such equations.While the usual Galerkin method can be interpreted as a projection of the considered equation on a linear space, the methods considered here are related to the projection of the equation on a nonlinear manifold. From the practical point of view some terms have been identified as small, and sometimes.(but not always) disregarded.

Introduction to Numerical Linear Algebra and Optimisation

Abstract: Preface Part I. Numerical Linear Algebra: 1. A summary of results on matrices 2. General results in the numerical analysis of matrices 3. Sources of problems in the numerical analysis of matrices 4. Direct methods for the solution of linear systems 5. Iterative methods for the solution of linear systems 6. Methods for the calculation of eigenvalues and eigenvectors Part II. Optimisation: 7. A review of differential calculus. Some applications 8. General results on optimisation. Some algorithms 9. Introduction to non-linear programming 10. Linear programming Bibliography and comments Main notations used Index.

Numerical Methods for Viscous Flows With Moving Boundaries

Abstract: A review of numerical algorithms for the analysis of viscous flows with moving interfaces is presented. The review is supplemented with a discussion of methods that have been introduced in the context of other classes of free boundary problems, but which can be generalized to viscous flows with moving interfaces. The available algorithms can be classified as Eulerian, Langrangian, and mixed, ie, Eulerian-Lagrangian. Eulerian algorithms consist of fixed grid methods, adaptive grid methods, mapping methods, and special methods. Langrangian algorithms consist of strictly Langrangian methods, Langrangian methods with rezoning, free Lagrangian methods and particle methods. Mixed methods rely on both Lagrangian and Eulerian concepts. The review consists of a description of the present state-of-the-art of each group of algorithms and their applications to a variety of problems. The existing methods are effective in dealing with small to medium interface deformations. For problems with medium to large deformations the methods produce results that are reasonable from a physical viewpoint; however, their accuracy is difficult to ascertain.

A uniformly accurate finite element method for the Reissner-Mindlin plate

Abstract: A simple finite element method for the Reissner–Mindlin plate model in the prim-itive variables is presented and analyzed The method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging It is proved that the method converges with optimal order uniformly with respect to thickness

A second order splitting method for the Cahn-Hilliard equation

Abstract: A semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.

A method for calculating stress intensities in bimaterial fracture

Abstract: A numerical method is presented for obtaining the values of K* 1,K * II and K* III in the elasticity solution at the tip of an interface crack in general states of stress. The basis of the method is an evaluation of theJ-integral by the virtual crack extension method. Individual stress intensities can then be obtained from further calculations ofJ perturbed by small increments of the stress intensity factors. The calculations are carried out by the finite element method but minimal extra computations are required compared to those for the boundary value problem. Very accurate results are presented for a crack in the bimaterial interface and compared with other methods of evaluating the stress intensity factors. In particular, a comparison is made with stress intensity factors obtained by computingJ by the virtual crack extension method but separating the modes by using the ratio of displacements on the crack surface. Both techniques work well with fine finite element meshes but the results suggest that the method that relies entirely on J-integral evaluations can be used to give reliable results for coarse meshes.

Numerical solution of large Lyapunov equations

Abstract: A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.

Optimal finite-element interpolation on curved domains

Abstract: This paper is devoted to a general theory of approximation of functions in finite-element spaces. In particular, the case is considered where the functions to be interpolated are on the one hand not very smooth, and on the other are defined on curved domains. Thus, a number of well-known optimal interpolation results are generalized.

Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media

Abstract: This paper presents a numerical method for simulating flow fields in a stochastic porous medium that satisfies locally the Darcy equation, and has each of its hydraulic parameters represented as one realization of a three-dimensional random field. These are generated by using the Turning Bands method. Our ultimate objective is to obtain statistically meaningful solutions in order to check and extend a series of approximate analytical results previously obtained by a spectral perturbation method (L. W. Gelhar and co-workers). We investigate the computational aspects of the problem in relation with stochastic concepts. The difficulty of the numerical problem arises from the random nature of the hydraulic conductivities, which implies that a very large discretized algebraic system must be solved. Indeed, a preliminary evaluation with the aid of scale analysis suggests that, in order to solve meaningful flow problems, the total number of nodes must be of the order of 106. This is due to the requirement that Δxi ≪ gli ≪ Li, where Δxi is the mesh size, λi is a typical correlation scale of the inputs, and Li is the size of the flow domain (i = 1, 2, 3). The optimum strategy for the solution of such a problem is discussed in relation with supercomputer capabilities. Briefly, the proposed discretization method is the seven-point finite differences scheme, and the proposed solution method is iterative, based on prior approximate factorization of the large coefficient matrix. Preliminary results obtained with grids on the order of one hundred thousand nodes are discussed for the case of steady saturated flow with highly variable, random conductivities.

A comparison of adaptive refinement techniques for elliptic problems

Abstract: Adaptive refinement has proved to be a useful tool for reducing the size of the linear system of equations obtained by discretizing partial differential equations. We consider techniques for the adaptive refinement of triangulations used with the finite element method with piecewise linear functions. Several such techniques that differ mainly in the method for dividing triangles and the method for indicating which triangles have the largest error have been developed. We describe four methods for dividing triangles and eight methods for indicating errors. Angle bounds for the triangle division methods are compared. All combinations of triangle divisions and error indicators are compared in a numerical experiment using a population of eight test problems with a variety of difficulties (peaks, boundary layers, singularities, etc.). The comparison is based on the L-infinity norm of the error versus the number of vertices. It is found that all of the methods produce asymptotically optimal grids and that the number of vertices needed for a given error rarely differs by more than a factor of two.

An efficient boundary element method for nonlinear water waves

Abstract: The paper presents a computational model for highly nonlinear 2-D water waves in which a high order Boundary Element Method is coupled with a high order explicit time stepping technique for the temporal evolution of the waves. The choice of the numerical procedures is justified from a review of the literature. Problems of the wave generation and absorption are investigated. The present method operates in the physical space and applications to four different wave problems are presented and discussed (space periodic wave propagation and breaking, solitary wave propagation, run-up and radiation, transient wave generation). Emphasis in the paper is given to describing the numerical methods used in the computation.

Digital holographic reconstruction of sources with arbitrarily shaped surfaces

Abstract: Nearfield acoustic holography has proven to be a useful tool for studying sound radiation. However, the analytic formulation and all current implementations of the technique require that the measurement and reconstruction surfaces be level surfaces of a separable coordinate system. In this article, a holographic process is presented based on numerical methods that work for source surfaces or measurement surfaces that may have an arbitrary shape.

A class of iterative methods for solving saddle point problems

Abstract: We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.

A fast `Monte-Carlo cross-validation' procedure for large least squares problems with noisy data

Abstract: We propose a fast Monte-Carlo algorithm for calculating reliable estimates of the trace of the influence matrixA ? involved in regularization of linear equations or data smoothing problems, where ? is the regularization or smoothing parameter. This general algorithm is simply as follows: i) generaten pseudo-random valuesw 1, ...,w n , from the standard normal distribution (wheren is the number of data points) and letw=(w 1, ...,w n ) T , ii) compute the residual vectorw?A ? w, iii) take the `normalized" inner-product (w T (w?A ? w))/(w T w) as an approximation to (1/n)tr(I?A ?). We show, both by theoretical bounds and by numerical simulations on some typical problems, that the expected relative precision of these estimates is very good whenn is large enough, and that they can be used in practice for the minimization with respect to ? of the well known Generalized Cross-Validation (GCV) function. This permits the use of the GCV method for choosing ? in any particular large-scale application, with only a similar amount of work as the standard residual method. Numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency.

A Weighted Average Flux Method for Hyperbolic Conservation Laws

Abstract: A numerical technique, called a 'weighted average flux' (WAF) method, for the solution of initial-value problems for hyperbolic conservation laws is presented. The intercell fluxes are defined by a weighted average through the complete structure of the solution of the relevant Riemann problem. The aim in this definition is the achievement of higher accuracy without the need for solving 'generalized' Riemann problems or adding an anti-diffusive term to a given first-order upwind method. Second-order accuracy is proved for a model equation in one space dimension; for nonlinear systems second-order accuracy is supported by numerical evidence. An oscillation-free formulation of the method is easily constructed for a model equation. Applications of the modified technique to scalar equations and nonlinear systems gives results of a quality comparable with those obtained by existing good high resolution methods. An advantage of the present method is its simplicity. It also has the potential for efficiency, because it is well suited to the use of approximations in the solution of the associated Riemann problem. Application of WAF to multidimensional problems is illustrated by the treatment using dimensional splitting of a simple model problem in two dimensions.

Numerical solution for response of beams with moving mass

Abstract: An analytical-numerical method is presented that can be used to determine the dynamic behavior of beams, with different boundary conditions, carrying a moving mass. This article demonstrates the transformation of a familiar governing equation into a new, solvable series of ordinary differential equations. The correctness of the results has been ascertained by a comparison, using finite element models, and very good agreement has been obtained. Furthermore, the article shows that the response of structures due to moving mass, which has often been neglected in the past, must be properly taken into account because it often differs significantly from the moving force model.

A relaxation procedure for domain decomposition methods using finite elements

Abstract: We present the convergence analysis of a new domain decomposition technique for finite element approximations. This technique was introduced in [11] and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent. No global preconditioner is needed in the practice, but simply single-domain finite element solvers are required. An optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated. Applications are given to both elliptic equations and incompressible Stokes equations.

Application of Cosserat theory in numerical solutions of limit load problems

Abstract: A continuum model for regular block structures is derived within the framework of Cosserat theory. On the basis of the derived model a limit load problem of tunnel statics is solved by the finite element method. In a special case the results of the numerical analysis are compared with a kinematical bounding solution.

Stokes Profile Analysis and Vector Magnetic Fields. II. Formal Numerical Solutions of the Stokes Transfer Equations

Abstract: Two numerical methods for formal integration of the Stokes transfer equations for line formation in a strong magnetic field were tested by computing Stokes profiles for a Zeeman triplet in a Milne-Eddington model atmosphere, and for the anomalously split Ca II K line in a realistic solar model. The first method is a Feautrier (1964) type method, in which the equations are written in second-order form and solved by finite-differences. The second method is a new solution called DELO, in which an integral equation for the Stokes vector is formulated in terms of the lambda operator (LO) associated with the diagonal elements (DE) of the absorption matrix. It is shown that the DELO method is faster and more accurate than the Feautrier method, and that both methods are more efficient than the Runge-Kutta integration method. 34 refs.

An historical note on finite rotations

Abstract: It is shown in this paper that Euler was first to derive the finite rotation formula which is often erroneously attributed to Rodrigues, while Rodrigues was responsible for the derivation of the composition formulae for successive finite rotations and the so-called Euler parameters of finite rotation. Therefore, based upon historical facts, the following nomenclature is suggested: Euler's finite rotation formula, Rodrigues' composition formulae of finite rotations, and Euler-Rodrigues parameters. The text of the paper contains modern symbols and formula forms, while the Appendices contain brief summaries from relevant historical sources with minor alterations in symbols at the most.

Numerical Analysis: A Comprehensive Introduction

Abstract: Systems of linear equations linear programming interpolation approximation of functions nonlinear equations Eigenvalue problems method of least squares numerical quadrature ordinary differential equations partial differential equations.